How to give particles systems a moving target (key)?
Simple solution
Setting the physics type to 'No' causes the particles to stick to their emitter, even if it moves. This can be used to make moving targets for other particle systems.
Before, I was using the &quot;Newtonian&quot; physics type, which caused the particles to stand still in the absence of any forces or initial velocity. I assumed 'No' physics would also result in stationary particles, but they actually stay stationary in the emitter reference frame. So yeah. That's it.
More complicated solution
Before realizing the simple solution exists. I did it by creating a 'middle' layer of keyed particle systems, which move and serve as targets for the rendered particle systems.
This is more complicated and taxes a computer significantly more, but leaving it here since it seems like it could be used to add field effects to the targets, which isn't possible with the 'No' physics setting from the simple solution.

The initial and final objects in the morph each have three, rather than one particle system.

The first particle system emits stationary particles at the beginning frame of the morph.
The second particle system emits stationary particles at the ending frame of the morph.
The third particle system has keyed physics and uses the first and second systems as targets. This causes it to move, tracing the motion of the emitter object (or at least going in a straight line between toward a destination, which is good enough to create the illusion).

This third, moving system (one in each mesh) is part of the 'middle' layer mentioned above in that it's used as a target for one of the rendered particle systems.
Here are the layers of particle systems:

Bottom layer: Stationary particle systems emitted by each mesh at the beginning and end of the morph.
Middle layer: For each mesh, one keyed particle system. For targets, this system uses the two stationary systems for its mesh (beginning and end). The particle systems in this middle layer are the moving targets the initial question is asking for.
Top layer: These are the rendered particle systems. Each of these systems is keyed and has two targets from the middle layer, one each for the pre- and post-morph meshes.

The resulting .blend file:

It has some additional complexity because each pre-morph mesh morphs to every post-morph mesh. Also, each morph has two rendered particle systems, one each for the pre- and post-morph mesh materials. In total, there are 96 particle systems! (Though this isn't the minimum necessary since the &quot;bottom layer&quot; systems are duplicated.)